Transcript Computer Simulations of Stratified Turbulence

Modeling Challenges and Approaches
in LES for Physically Complex Flows
J. Andrzej Domaradzki
Peter Diamessis
Xiaolong Yang
Department of Aerospace and Mechanical Engineering
University of Southern California
Los Angeles
Financial support: NSF and ONR
Introduction
Classical LES equations for a constant
density, incompressible flow:
u i

1 p
 ui


(ui u j )  
 2 
 ij
t x j
 xi
x j x j
2
Complexity sources in LES modeling:
• GEOMETRY
• PHYSICS: governing equations have additional
terms
“Complex” Physics
• Compressibility
• Rotation and Stratification (Stable/Unstable)
u i

1 p
 2 ui


(u j ui )  
 2 
 ij
t x j
 xi
x j x j
2 j ijl ul  gi T
Additional terms in the momentum equation are
linear and do not require modeling
• Temperature Equation
T

 T


(u j T )  

 j
2
t
x j
x j
x j
2
• Subgrid Scale Stresses
 ij  ui u j  ui u j
Same form as for a flow
without Coriolis force
  j  u jT  u j T
Same form as for a
passive scalar
• Can/should traditional models be used?
Incompressible MHD equations
ui

1 p
 ui


(u j ui )  
 2 
 ij 
t x j
 xi
x j x j
2

 b
1
ext 

(b j bi ) 
 ij 
Bj
bi
(  ) x j
x j
(  )
x j
1
bi

 bi
ext 

(b j ui  u j bi )  B j
ui   2
t x j
x j
x j
2
Turbulence MHD (Rm << 1) – from R. Moreau
Acquis
Turbulence
homogène
Ecoulement de
Hartmann
Ecoulements
complexes
Défis
Pas de cascade !
1
 t 2
l//

Anisotropie:
l   J 
Spectre d’énergie : E  k3t2
Début des DNS et LES
Transferts angulaires
Quid de u//
?
u
Quid des petites échelles ?
Quid du 2ème scalaire ?
LES spécifique à la MHD ?
Régime Q2D intense
Casc. inverse d’énergie
Le tourbillon Q2D MHD
LES et RANS
Fonctions de paroi
Rôle des couches de Ha
Promoteurs de turbulence ?
RIEN !
Profils de vitesse en M
Entrée et sortie de l’aimant
Géométries complexes
(divergents, coudes, etc…)
Rotating Turbulence
• Rotating turbulence: refers to flows observed in a frame of
reference rotating with a solid body angular velocity .
• Rotating flows are distinguished from ‘non-rotating’ flows by the
presence of the Coriolis force (turbomachinery, geophysical flows).
• Rossby number: Ro  U / L
for turbulent rotating flow: Ro   /(2 K )
• Qualitative Observations:
* energy decay is reduced compared with non-rotating turbulence
* the inertial spectrum is steeper than the Kolmogoroff k-5/3 form
* for initially isotropic flow Reynolds stress remains isotropic but
length scales become anisotropic
Modeling Difficulties
Implications for SGS models
but
CS*  CS
makes the model too dissipative for rotating turbulence
• For dynamic model
CS*  CS
and the model is inconsistent with the
transformation properties
Approximate velocity models (nonlinear, deconvolution, estimation) avoid these
difficulties and satisfy transformation properties automatically
Truncated Navier-Stokes Equations (TNS)
• Variation of the Velocity Estimation Model (VEM)
(Domaradzki and Saiki (1997))
• Based on two observations:
- the dynamics of small scales are strongly determined by
the large, energy carrying eddies
- the contribution of small scales to the dynamics of large
scales (k<kc) comes mostly from scales within kc<k<2kc
• Implemented for low Reynolds number rotating
turbulence by Domaradzki and Horiuti (2001) to avoid
difficulties with rotational transformation properties for
classical SGS models (Horiuti (2001))
Truncated N-S dynamics (spectral space)
Estimated scales (on “fine” mesh):
Artificial energy accumulation due
to absence of (natural or eddy)
viscosity.
E(k)
Large physical
scales (on
“coarse” mesh):
computed by N-S
eqns.
Unresolved
scales
kc
Filter small-scales at fixed interval and
replenish using estimation model
2kc
k
TNS=Sequence of DNS runs with
periodic processing of high modes
Multiscale modeling
Scales periodically replaced
by estimated scales
E(k)
Large scales
computed from
(inviscid) TNS eqs.
Unresolved
scales
kc
2kc
Estimated small scales computed from
a separate dissipative equation forced
by the inviscid solution.
k
Similar to Dubrulle,
Laval, Nazarenko,
Kevlahan (2001)
Properties
• N-S equations are solved
- SGS stresses are not needed
- Transformation properties (Galilean, rotating frame) always satisfied
- Commutation errors are avoided
• Applicable to strongly anisotropic flows (VLES)
• Straightforward inclusion of additional effects (convection, compressibility,
stable stratification, rotation)
• Requires determination of the filtering interval (based on a small eddy
turnover time or a limiter on the small energy growth)
TNS for rotating turbulence
• TNS with VEP applied to simulate low/high Re number
turbulence with/without rotation
• DNS data of Horiuti (2001)
• Mesh size is 2563 for DNS and 643 for TNS
• The initial condition for TNS is obtained by truncating the
full 2563 DNS field to 323 grid
• Low Re:   0.0014, tmax  1.5
• High Re:   2.5 1016 , tmax  20
Energy Spectrum, low Re
0
 10
Energy decay, high Re
-1.2
High Re: spectral slope predictions k
n
• n=2: Zhou (1995); Baroud et al. (2002).
• n=11/5=2.2: Zeman (1994).
• n=7/3=2.33: Bershadskii, Kit, Tsinober
(1993).
• n=3: Smith and Waleffe (1999); Cambon et
al. (2003).
Energy spectrum, high Re
-2
-3
Anisotropy Indicators
• Length scales
L   
,

0
u ( x )u ( x  rn ) dr
u ( x )u ( x )
• Reynolds stress tensor and anisotropy
tensor
Rij  ui ( x )u j ( x )   Re Uˆ ij (k )  dk
bij  Rij / q 2   ij / 3
(=0 for isotropic turbulence)
Integral length scales
5,10
50
1
100
0
Anisotropy Tensor
• Directional and polarization anisotropy tensor
bij  bije  bijz

bije   e(k )  E (k )
4 k 2

Pij dk / q 2
bijz   Re  ZNi N j  dk / q 2
E(k) is the total energy for all modes in a
wavenumber shell k | k |
Directional anisotropy tensor
5
100
1
0
Summary of Observations
• Spectral slope n=-2 at earlier times (t<5) and n=-3 at later
times (t>15)
• Anisotropy indicators largest for
- times t>5
- moderate rotation rates
• Anisotropy indicators small for   0 and   
Spectral Exponent Hypothesis
 Approximately isotropic state characterized by n=-2
 Strongly anisotropic state characterized by n=-3
Two different views of LES
• Classical view:
- governing LES equations are derived from NavierStokes eqs. and are are different from them
- unknown SGS stress is modeled using physical
principles
- there exists a unique best solution to the SGS modeling
problem
Additional “complex” physics often requires substantial
changes in models developed for simpler flows.
• Competing view:
- governing LES equations are simply Navier-Stokes eqs.
- LES modeling problem is of numerical nature: how to accurately
solve Navier-Stokes eqs. on coarse grids
- there may be many solutions to the problem, e.g. regularization of
the equations or the solutions, using numerical dissipation in place
of physical dissipation (MILES/ILES), etc.
Disadvantages: ILES is not robust because there is no
guarantee that the implicit dissipation is equal to the
physical dissipation
Potential Advantage: if the equations are known there are
no modeling problems!
Turbulent wakes in stably stratified fluids
Guadalupe island
D = 10 m
U = 10 m/s
N = 0.003 /s
Re = 108
F = 500
D = 10 km
U = 10 m/s
N = 10-4 /s
Re = 1010
F = 10
Experiments: Spedding et al. (1996, 1997,2001,2002).
(a)
T
D
H

U
(z)
H
 20
D
Re 
UD

2U
F
ND
 103 , 10 4 
 1, 200
Numerical Method: Computational Domain
and Flow Configuration
• Periodic in horizontal directions: Fourier discretization.
• Bottom: Solid Wall. Top: Free Surface.
Divide into spectral subdomains (elements).
Legendre polynomial discretization.
Wake of a towed sphere
Numerical Method:
Spectral Multidomain Discretization
Well-resolved
wake core,
subsurface
Ambient region
not over-resolved
• Partition domain into M subdomains with:
– Height Hk and order polynomial approximation Nk.
– Non-uniform local Gauss-Lobatto grid (No stretching coefficients !).
Numerical Techniques Dealing with UnderResolution to Maintain Spectral Accuracy
and Stability
•
•
•
Spectral Filtering.
Strong Adaptive Interfacial Averaging.
Spectral Penalty Methods (J. Hesthaven – SIAM J. Sci.
Comp. Trilogy)
Attempting to satisfy eqs. with limited resolution arbitrarily close to
boundaries leads to catastrophic instabilities
Solution: Implement BC in a weak form by collocating equation at
boundary with a penalty term
ui
 RHS   BC
t
Truncated Navier-Stokes Dynamics
Flow Parameters and Runs Performed
Re=UD/ Fr=2U/ND
Pr=/
Resolution
Nk
5x103

1
128x128x165
32
5x103
4
1
-””-
-””-
2x104

1
128x128x249
44
2x104
4
1
-””-
-””-
• Domain size: 16Dx16Dx12D -Timestep Dt~0.03 D/U.
• Initialization procedure that of Dommermuth et al. (JFM 2002) =
Relaxation.
Initial velocity data that of Spedding at Nt=3.
Flow Structure: Isosurfaces of |ω| at Fr= 
Re=5K
Re=20K
Flow Structure: Isosurfaces of ωz at Fr=4
Re=5K
Re=20K
Vertical Vorticity, wz at Horizontal Centerplane
(Nt=56, x/D=112)
Re=5K
Fr=4
Fr= 
Re=20K
Fr and Re Universality of Wake Power Laws: Mean
centerline velocity
Fr and Re Universality of Wake Power Laws:
Wake Horizontal Lengthscale
, Fr=4
Conclusions
• A range of subgrid scales adjacent to the
resolved range dominates dynamics of the
resolved eddies
• These subgrid scales can be estimated in terms
of the resolved scales (estimation model)
• Dynamics of the resolved eddies is
approximated by Truncated Navier-Stokes
equations for resolved and estimated scales
• The method consists of a sequence of
underresolved DNS and a periodic processing of
the solution
Conclusions
• TNS approach captures well temporal evolution of wake mean
velocity profile, length scales and vorticity field structure.
• For Reynolds numbers considered both TNS and stability
filtering produced essentially the same results (supports ILES?)
• For decaying isotropic turbulence the inertial range spectrum is
maintained during flow evolution
• For decaying rotating turbulence good comparison with DNS
data is obtained at low Re
• At high Re decreased kinetic energy decay rates are observed
for increasing rotation rate and the asymptotic spectrum
proportional to k 3
Can LES with complex physics be best addressed
by minimizing explicit modeling that affects the
form and properties of the governing equations !?